Decoding for algebraic geometric code associated with a fiber product

Abstract

The present invention concerns a method of decoding a one-point algebraic geometric code defined on an algebraic curve represented by an equation in X and Z of degree 2^μφ in Z, where φ is a strictly positive integer and μ an integer greater than 1, obtained by taking the fiber product of μ component algebraic equations, each of said component equations governing the unknown X and an unknown Y_i, where i=0, . . . , μ−1, and being of degree 2^φ in Y_i. This method comprises the decoding of 2^(μ−1)φ“clustered” codes, all defined on the same algebraic curve represented by one of said component equations. The invention also relates to devices and apparatuses adapted to implement this method.

Description

[0001] This application claims priority from French Patent Application No. 04 00 267 filed on Jan. 13, 2004, which is incorporated herein by reference. FIELD OF THE INVENTION [0002] The present invention concerns systems for communication or recording of data in which the data are subjected to a channel encoding in order to improve the fidelity of the transmission or storage. It concerns more particularly a decoding method, as well as the devices and apparatus adapted to implement this method. BACKGROUND OF THE INVENTION [0003] It will be recalled that channel “block encoding” consists, when the “codewords” sent to a receiver or recorded on a data carrier are formed, of introducing a certain level of redundancy in the data. More particularly, by means of each codeword, the information is transmitted that is initially contained in a predetermined number k of symbols taken from an “alphabet” of finite size q; on the basis of these k information symbols, calculation is made of a number...

AI Technical Summary

Benefits of technology

[0021] However, the authors of the present invention have discovered that it is possible to reconcile the apparently contradictory properties of high genus g and low decoding complexity, where the algebraic curve is of a particular type belonging to the general class of “fiber products”, which will now be defined.

Problems solved by technology

There is a transmission error if the difference e between a received word r and the corresponding codeword v sent by the transmitter is non-zero.

However, in all cases, the concern is not with a possibility in principle, since it is often difficult to develop a decoding algorithm achieving such performance.

When such a carrier is affected by a physical defect such as a scratch, a high number of information symbols may be rendered unreadable.

Consequently, if a Reed-Solomon code is desired having codewords of great length, high values of q must be envisaged, which leads to costly implementations in terms of calculation and storage in memory.

Algebraic geometric codes are advantageous as to their minimum distance d, which is at least equal to (n−k+1−g), and, as has been said, as to the length of the codewords, but they have the drawback of requiring decoding algorithms that are rather complex, and thus rather expensive in terms of equipment (software and / or hardware) and processing time.

Generally, the higher the genus g of the algebraic curve used, the greater the length of the codewords, but also the greater the complexity of the decoding.

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